Building Models of Determinacy from Below

TL;DR

This paper introduces a novel construction method to produce minimal models of certain determinacy axioms, expanding the understanding of models where no hod mouse has a measurable limit of Woodins.

Contribution

It develops an $L$-like construction capable of generating minimal models of $ ext{AD}_ ext{R}+$ with specific properties, including models of $ ext{AD}^++ ext{AD}_ ext{R}+V=L(P( ext{R}))$ without certain hod mice.

Findings

Constructs minimal models of $ ext{AD}_ ext{R}+$ with $ heta$ regular.

Can produce models of $ ext{AD}^++ ext{AD}_ ext{R}+V=L(P( ext{R}))$ without hod mice with measurable limits of Woodins.

Provides a framework for analyzing models of determinacy with specific structural properties.

Abstract

We present an $L$-like construction that produces the minimal model of $\mathsf{AD}_\mathbb{R}+$"$\Theta$ is regular". In fact, our construction can produce any model of $\mathsf{AD}^++\mathsf{AD}_\mathbb{R}+V=L(P(\mathbb{R}))$ in which there is no hod mouse with a measurable limit of Woodins.